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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 56784bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
56784.bh6 | 56784bk1 | \([0, -1, 0, 2648, 10672]\) | \(103823/63\) | \(-1245548408832\) | \([2]\) | \(73728\) | \(1.0101\) | \(\Gamma_0(N)\)-optimal |
56784.bh5 | 56784bk2 | \([0, -1, 0, -10872, 97200]\) | \(7189057/3969\) | \(78469549756416\) | \([2, 2]\) | \(147456\) | \(1.3567\) | |
56784.bh3 | 56784bk3 | \([0, -1, 0, -105512, -13076688]\) | \(6570725617/45927\) | \(908004790038528\) | \([2]\) | \(294912\) | \(1.7033\) | |
56784.bh2 | 56784bk4 | \([0, -1, 0, -132552, 18592560]\) | \(13027640977/21609\) | \(427223104229376\) | \([2, 2]\) | \(294912\) | \(1.7033\) | |
56784.bh4 | 56784bk5 | \([0, -1, 0, -91992, 30144048]\) | \(-4354703137/17294403\) | \(-341920891084910592\) | \([2]\) | \(589824\) | \(2.0498\) | |
56784.bh1 | 56784bk6 | \([0, -1, 0, -2119992, 1188797232]\) | \(53297461115137/147\) | \(2906279620608\) | \([2]\) | \(589824\) | \(2.0498\) |
Rank
sage: E.rank()
The elliptic curves in class 56784bk have rank \(0\).
Complex multiplication
The elliptic curves in class 56784bk do not have complex multiplication.Modular form 56784.2.a.bk
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.